What are mountain waves?
Many types of atmospheric motions can be described as waves. For example, Rossby waves describe the large scale motion of the atmosphere, and have wavelengths on the order of thousands of kilometers. However, there are also many small scale waves found in the atmosphere with much shorter wavelengths. One example of these are gravity waves, which are also referred to as buoyancy waves.
Every wave has a restoring force, which forces the disturbance back towards its equilibrium point. For Rossby waves, this restoring force is the Coriolis force, which is an apparent force generated by the rotation of the Earth. Another example is sound waves, where the restoring force is generated by pressure perturbations. For gravity waves, the restoring force is buoyancy, which is indirectly related to gravity. This is where the common name "gravity waves" originates from, although it is technically inaccurate.
Mountain waves are a special case of atmospheric gravity waves. They are internal waves (as opposed to external waves found at the interface between two fluids, like the surface of water). Internal waves arise in a stable and continuously stratified fluid. (See Gill (1982) for a more lengthy and quantitative explanation.)
In order for waves to form, there needs to be an initial source of the disturbance. Mountain waves are generated by air that is forced to flow over raised topography. The lift by the mountain sets up an oscillation downstream of and above the peak. This is the mountain wave.
Why do they matter?
These waves are important for several reasons. The most visible impact of mountain waves is on aviation. It is no coincidence that a large portion of turbulence felt in flight occurs over mountains. Much of the turbulence found above mountain ranges are due to the strong vertical air motions found in mountain waves.
In addition, it has been shown (see Eliassen and Palm (1960)) that mountain waves have a direct impact on the general circulation of the atmosphere. The waves transport the drag force exerted by the mountain upward. When they break, they "deposit" this drag force and slow down the large-scale flow.
This effect can be quite significant. Bretherton (1969) showed that this "gravity wave drag (GWD)" led to large model errors when it was not accounted for in weather prediction models. Palmer (1987) implemented one of the first GWD parameterizations and achieved a significant improvement in forecast accuracy. Ever since, continued study of mountain waves and development of parameterizations have been an considerable part of model development.